Rationalize the denominatior of :
`(2-sqrt(3))/(2+sqrt(3))`

Answer

To rationalize the denominator of the expression \((2 - \sqrt{3}) / (2 + \sqrt{3})\), we can follow these steps:

### Step 1: Identify the expression
We start with the expression:
\[ \frac{2 - \sqrt{3}}{2 + \sqrt{3}} \]

### Step 2: Multiply by the conjugate
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is \(2 - \sqrt{3}\):
\[ \frac{(2 - \sqrt{3})(2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})} \]

### Step 3: Expand the numerator
Now, we need to expand the numerator:
\[ (2 - \sqrt{3})(2 - \sqrt{3}) = 2^2 - 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2 \]
Calculating this gives:
\[ = 4 - 4\sqrt{3} + 3 = 7 - 4\sqrt{3} \]

### Step 4: Expand the denominator
Next, we expand the denominator using the difference of squares:
\[ (2 + \sqrt{3})(2 - \sqrt{3}) = 2^2 - (\sqrt{3})^2 \]
Calculating this gives:
\[ = 4 - 3 = 1 \]

### Step 5: Combine the results
Now we can combine the results from the numerator and denominator:
\[ \frac{7 - 4\sqrt{3}}{1} = 7 - 4\sqrt{3} \]

### Final Answer
Thus, the rationalized form of the expression \(\frac{2 - \sqrt{3}}{2 + \sqrt{3}}\) is:
\[ 7 - 4\sqrt{3} \]

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